Functional data objects derived from high-frequency financial data often exhibit volatility clustering. Versions of functional generalized autoregressive conditionally heteroscedastic (FGARCH) models have recently been proposed to describe such data, however so far basic diagnostic tests for these models are not available. We propose two portmanteau type tests to measure conditional heteroscedasticity in the squares of asset return curves. A complete asymptotic theory is provided for each test. We also show how such tests can be adapted and applied to model residuals to evaluate adequacy, and inform order selection, of FGARCH models. Simulation results show that both tests have good size and power to detect conditional heteroscedasticity and model mis-specification in finite samples. In an application, the tests show that intra-day asset return curves exhibit conditional heteroscedasticity. This conditional heteroscedasticity cannot be explained by the magnitude of inter-daily returns alone, but it can be adequately modeled by an FGARCH(1,1) model.

中文翻译：

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测试函数数据的条件异方差性

源自高频金融数据的功能数据对象通常表现出波动性聚类。最近有人提出了功能广义自回归条件异方差 (FGARCH) 模型的版本来描述此类数据，但是到目前为止，这些模型的基本诊断测试尚不可用。我们提出了两个混合类型检验来衡量资产收益曲线平方的条件异方差性。每个测试都提供了完整的渐近理论。我们还展示了如何调整此类测试并将其应用于模型残差以评估 FGARCH 模型的充分性并为订单选择提供信息。仿真结果表明，这两种检验都具有良好的规模和能力，可以检测有限样本中的条件异方差和模型指定错误。在一个应用程序中，检验表明，日内资产收益率曲线表现出条件异方差性。这种条件异方差不能仅用日间收益的大小来解释，但可以通过 FGARCH(1,1) 模型充分建模。